BEST-FIT STRAIN FROM MULTIPLE ANGLES OF SHEAR AND IMPLEMENTATION IN A COMPUTER PROGRAM FOR GEOLOGICAL STRAIN ANALYSIS

An analytic solution to the Wellman (1962) construction was desired for a 2D and 3D strain analysis program, EllipseFit 2, designed for field-based strain studies and structural geology labs. Oriented images are digitized using center-points, five-point ellipses, polygon-moment ellipses, or line segment pairs. Techniques include center-to-center (Fry, 1979; Erslev, 1988) with ellipse-fitting, Rf/Φ (Dunnet, 1969) and polar (Elliott, 1970) graphs. Best-fit ellipse calculations include shape-matrix eigenvalues (Shimamoto and Ikeda, 1976), mean radial length (Mulchrone et al., 2003), and hyperboloidal vector mean (Yamaji, 2008), with error analysis. Fitting of section-ellipses to ellipsoids is done using Shan's (2008) method.

A Wellman construction (1962) is used to determine strain from pairs of initially orthogonal lines. Ragan and Groshong (1993) gave a trigonometric solution for two angles of shear, however the tangent function makes this numerically unstable, so an analytic geometry solution was derived. For two line segments, p with endpoints (x₀, y₀)^T and (x₁, y₁)^T, and q with endpoints (x₂, y₂)^T and (x₃, y₃)^T, p and q are translated: p' = p - (x₀ + 1, y₀)^T, q' = q - (x₂ - 1 , y₂)^T so one endpoint of p lies at (-1, 0)^T, and one endpoint of q lies at (1, 0)^T. The implicit forms for the two translated lines are: a₁x + a₂y + a₃ = 0, b₁x + b₂y + b₃ = 0, where: a = (y₀' – y₁', x₁' – x₀', x₀'y₁' – x₁'y₀')^T, b = (y₂' – y₃', x₃' – x₂', x₂'y₃' – x₃'y₂')^T. Solving these gives two diametrically opposed points x, y and -x, -y on the ellipse: x = (b₁c₂ – b₂c₁) / (a₁b₂ – a₂b₁), y = (a₂c₁ – a₁c₂) / (a₁b₂ – a₂b₁). For two pairs of line segments the strain can be solved directly. For two or more pairs the best-fit ellipse can be found by a least-squares minimization directly or by a LU decomposition to solve for the coefficients of the quadratic equation of a centered ellipse.